This thesis discusses the two dimensional allotrope of carbon known as graphene in presence of magnetic field, with special focus on edge states. The structure of graphene is described in detail and from the structure, two models are formed. The Dirac equation is a good description of graphene for large samples, far away from edges, where the boundaries can be ignored. However, it causes problems with most types of edge and hard wall approximation has to be implemented.
The Dirac equation is described in detail and used to obtain an energy spectrum, wavefunction and density of states for graphene edge in a strong magnetic field. For comparison, a Bohr-Sommerfield approximation was used to find the dispersion relation and compare it to the results obtained numerically from the Dirac equation.
The second model, better fitting for nano-scale systems, is the tight binding model. This model was utilized to find Energy spectrum for graphene flakes in magnetic field, which resembles Hofstadter's butterfly spectrum. The spectrum was analyzed and periodic oscillations of magnetisation dependent on magnetic field (known as the de Haas-van Alphen effect) were described. The oscillation of magnetisation depends on the shape of the dot, even though the main properties remain the same: at low magnetic field, periodic oscillations due to Aharonov-Bohm effect, turning into more chaotic oscillations depending on the boundary conditions of the given quantum dot.

Description:

A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of Doctor of Philosophy of Loughborough University.